where \(\phi _N(E,0)\)  is the primary
flux of nucleons in the atmosphere, \(m_H\)  is the decaying particle's mass, \(\tau _H\)  is the proper
lifetime of the hadron, \(h_0 = 6.4\)  km,
\(f(\theta ) \approx 1/\cos \theta \)  for \(\theta < 60^o\) , and the effective
interaction lengths \(\Lambda _i\)  are given by \(\Lambda _i = \lambda _i/(1 -Z_{ii})\) , with \(\lambda _i\)  being the associated interaction length (\(i =N,H\) ). For \(Z_{H\nu }\) , our treatment of the semileptonic decay of \(D\) -hadrons follows closely Ref. [1]}. In particular, we assume the analytical decay distributions \(H \rightarrow \mu \nu _{\mu }X\)  obtained in Ref. [2]} and use the decay branching ratios reported in the most recent PDG [3]}.
For a detailed discussion of the cascade equations, see
e.g. Refs. [4]}, [5]}.
Assuming that the incident flux can be represented by protons (\(N = p\) ), the charmed
hadron \(Z\) -moments are given by
\(Z_{pH} (E) = \int _0^1 \frac{dx_F}{x_F} \frac{\phi _p(E/x_F)}{\phi _p(E)}\frac{1}{\sigma _{pA}(E)} \frac{d\sigma _{pA \rightarrow H}(E/x_F)}{dx_F} \,\,,\) 
